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Article  |   October 2019
Contributions of foveal and non-foveal retina to the human eye's focusing response
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Journal of Vision October 2019, Vol.19, 18. doi:https://doi.org/10.1167/19.12.18
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      Vivek Labhishetty, Steven A. Cholewiak, Martin S. Banks; Contributions of foveal and non-foveal retina to the human eye's focusing response. Journal of Vision 2019;19(12):18. doi: https://doi.org/10.1167/19.12.18.

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Abstract

The human eye changes focus—accommodates—to minimize blur in the retinal image. Previous work has shown that stimulation of nonfoveal retina can produce accommodative responses when no competing stimulus is presented to the fovea. In everyday situations it is very common for the fovea and other parts of the retina to be stimulated simultaneously. We examined this situation by asking how nonfoveal retina contributes to accommodation when the fovea is also stimulated. There were three experimental conditions. (a) Real change in which stimuli of different sizes, centered on the fovea, were presented at different optical distances. Accommodation was, as expected, robust because there was no conflicting stimulation of other parts of the retina. (b) Simulated change, no conflict in which stimuli of different sizes, again centered on the fovea, were presented at different simulated distances using rendered chromatic blur. Accommodation was robust in this condition because there was no conflict between the central and peripheral stimuli. (c) Simulated change, conflict in which a central disk (of different diameters) was presented along with an abutting peripheral annulus. The disk and annulus underwent opposite changes in simulated distance. Here we observed a surprisingly consistent effect of the peripheral annulus. For example, when the diameter of the central stimulus was 8° (thereby stimulating the fovea and parafovea), the abutting peripheral annulus had a significant effect on accommodation. We discuss how these results may help us understand other situations in which nonfixated targets affect the ability to focus on a fixated target. We also discuss potential implications for the development of myopia and for foveated rendering.

Introduction
Fixating an object involves two oculomotor functions: (a) Rotating the eyes so that images of the object fall on the two eyes' foveas, and (b) changing the power of the crystalline lenses so that the retinal images are sharp. The latter is accommodation. By moving the eyes and focusing the lenses, one can see finer detail because the fovea supports much higher resolution than the rest of the retina and because detail is more visible in sharp images than in blurred images. It is often assumed, therefore, that stimuli on the fovea are the primary driver of accommodation in humans (Campbell, 1954; Fincham, 1951; Phillips, 1974). We investigate that assumption here. 
Retinal regions
The human retina is distinctly nonuniform. In the center is the macula, which is subdivided into concentric areas with progressively lower densities of photoreceptors and therefore progressively poorer visual resolution: the foveola (central 1°), fovea (outer diameter of 5°), parafovea (8°), and perifovea (17°; Provis, Dubis, Maddess, & Carroll, 2013). These regions are schematized in Figure 1. Beyond the macula is the peripheral retina that consists of the majority of the retinal surface. Here we examine how stimulation of various parts of the macula affects human accommodation. We will refer to central stimuli and peripheral stimuli, wherein the former specifies a disk centered on the foveola and the latter on a larger annulus that abuts the disk. 
Figure 1
 
Stimulation of foveal and nonfoveal retina. A smart phone is held at a distance of 50 cm and the viewer fixates the center of it. The concentric rings represent the foveola, fovea, parafovea, and perifovea, and how they would be stimulated in this situation. The perifovea is stimulated by more distant points than the fovea.
Figure 1
 
Stimulation of foveal and nonfoveal retina. A smart phone is held at a distance of 50 cm and the viewer fixates the center of it. The concentric rings represent the foveola, fovea, parafovea, and perifovea, and how they would be stimulated in this situation. The perifovea is stimulated by more distant points than the fovea.
Retinal eccentricity and accommodation
There is experimental and clinical evidence suggesting that nonfoveal stimulation can drive accommodation. For example, Hartwig, Charman, and Radhakrishnan (2011) measured accommodative responses when the parafovea or perifovea was stimulated but the fovea was not. Subjects fixated the middle of a blank central field and accommodative stimuli were presented at retinal eccentricities of 5°–15°. They observed significant responses at 5° and smaller, but also consistent responses at 10° and 15°. Similarly, Gu and Legge (1987) presented accommodative stimuli to different parts of the retina when subjects fixated a small target that provided no usable stimulus to accommodation. They found clear responses at retinal eccentricities of 1°–7.5° and smaller, but consistent responses at 15°–30°. Others have confirmed these observations (Bullimore & Gilmartin, 1987; Hennessy & Leibowitz, 1971; Hung & Ciuffreda, 1992; Phillips, 1974; Semmlow & Tinor, 1978). Thus, there is good evidence that stimulation of nonfoveal retina can drive accommodation when there is no informative stimulus at the fovea. 
It is also interesting that people with central-field loss retain some ability to accommodate. For example, patients with juvenile macular degeneration (loss of foveal function beginning early in life) exhibit accommodative responses to stimuli presented monocularly (White & Wick, 1995). This too suggests that nonfoveal stimulation can be effective when there is no usable foveal stimulus. 
Peripheral focus and myopia
Myopia, commonly referred to as near-sightedness, is a refractive condition in which light rays from a distant object are focused in front of the retina. This produces a blurred retinal image for those objects, which is usually corrected with spectacles or contact lenses. The prevalence of myopia has increased dramatically, especially in East and South Asia (Morgan, He, & Rose, 2017; Pan, Ramamurthy, & Saw, 2012; Wong & Saw, 2016). For example, 97% of 19-year-old males in Seoul (Jung, Lee, Kakizaki, & Jee, 2012) and 96% of university students in Shanghai (Sun et al., 2012) are now myopic, much more than a few decades earlier. Similar dramatic increases have occurred in other parts of the world (Dayan et al., 2005; Vitale, Sperduto, & Ferris, 2009). If the current trend continues, five billion people will be myopic by 2050 (Holden et al., 2016). 
Myopia is a significant risk factor for a variety of eye diseases later in life, including retinal detachment (Crim et al., 2017), cataract (Jeon & Kim, 2011), and glaucoma (Morita, Funata, & Tokoro, 1995). For this reason, there is strong motivation for understanding what causes myopia and for developing preventative measures. 
Several researchers and clinicians have hypothesized that a child's visual environment is an important contributor to the development of myopia: in particular, an increase in near work and/or an increase in exposure to indoor lighting (Morgan et al., 2018; Read, Collins, & Vincent, 2014; Zylbermann, Landau, & Berson, 1993). The near-work hypothesis states that activities, such as viewing a mobile device at close range, forces the child to focus near (by increasing curvature of the crystalline lens) and that this effort promotes eye growth, which in turn produces myopia (Morgan et al., 2018; Zylbermann et al., 1993). 
Studies in non-human infant primates have shown that where the image is habitually focused relative to the peripheral retina is an important predictor of myopic progression (Smith III, Hung, & Huang, 2009; Smith, Kee, Ramamirtham, Qiao-Grider, & Hung, 2005; Wallman & Winawer, 2004). When the image is generally focused behind the retina in the periphery, the developing eye lengthens, thereby moving the peripheral retina toward the position of best focus. Interestingly, the lengthening occurs even when the image at the fovea is habitually focused on that part of the retina. Thus, as the eye lengthens, myopia is produced at the fovea. These findings in non-human primates imply that where the image is formed in the periphery is important to determining whether the eye develops myopia or not, but there is no clear evidence for or against this hypothesis in developing humans. 
Current study
As we have seen, there is evidence that stimulation of nonfoveal retina affects accommodation when there is no usable foveal stimulus. But in everyday situations, like the one depicted in Figure 1, the fovea and nonfoveal regions are essentially always stimulated, so the previous work does not inform us about the contribution of nonfoveal retina under typical viewing conditions. We conducted an experiment to determine what that contribution is. 
Methods
Subjects
Eleven naive subjects (aged 18–30 years) participated: nine women and two men. Six were myopes (−1.0 to −8.0 D) and five were emmetropes. The myopes were habitual contact-lens wearers and wore their lenses during the study. Subjects with amblyopia, anisometropia, or strabismus were not included. The experiment was approved by the institutional ethical review board at University of California, Berkeley, and conducted in accordance with the Declaration of Helsinki. 
Apparatus
The apparatus is schematized in Figure 2. A detailed description is provided in Cholewiak, Love, and Banks (2018). The subject's left eye was stimulated while the right eye's accommodative response was measured. Accommodation is yoked between the two eyes so measuring one eye while stimulating the other is justified (Campbell, 1960). The subject's head was stabilized with an adjustable chin-and-forehead rest. Stimuli were projected onto a screen by a DLP projector (Texas Instruments LightCrafter 4710; Texas Instruments, Dallas, TX). The room was otherwise dark. Screen resolution was 1920 × 1080. The R, G, and B primaries were LEDs with relatively narrow spectra that were further narrowed using a triple-bandpass filter (Chroma 69002m; Chroma Technology, Bellows Falls, VT). The screen was 1.28 m from the subject's eye and subtended 32° × 18°. A focus-adjustable lens (Optotune EL-10-30-TC; Optotune, Dietikon, Switzerland) was placed just in front of the left eye. This lens was used to adjust the focal distance of the stimulus. The lens had almost no longitudinal chromatic aberration (Abbe number = 100). A −15.9 diopter (D) achromatic offset lens (Comar 63 DN 25; Comar Optics, Linton, Cambridgeshire, UK) was placed in the optical path to provide a range of focal distances of −7.6 to +4.1 D. A physical aperture, restricting the total field of view to 14°, was located 14.8 D (6.7 cm) from the eye, so its image was very blurry and therefore did not create an effective stimulus to accommodation. 
Figure 2
 
Experimental apparatus. Stimuli were projected from a DLP projector through a triple-bandpass filter onto a screen. The stimuli were viewed by the subject's left eye through a focus-adjustable lens, fixed offset lens, and aperture. The right eye's refractive state was measured by an autorefractor. Infrared light was shone into that eye and the vergence of its reflection from the retina was analyzed. A diffusing screen scattered the visible light from the experimental stimulus creating a uniform field seen by the right eye. A hot mirror enabled transmission of visible light to the two eyes and reflection of infrared light to the right eye.
Figure 2
 
Experimental apparatus. Stimuli were projected from a DLP projector through a triple-bandpass filter onto a screen. The stimuli were viewed by the subject's left eye through a focus-adjustable lens, fixed offset lens, and aperture. The right eye's refractive state was measured by an autorefractor. Infrared light was shone into that eye and the vergence of its reflection from the retina was analyzed. A diffusing screen scattered the visible light from the experimental stimulus creating a uniform field seen by the right eye. A hot mirror enabled transmission of visible light to the two eyes and reflection of infrared light to the right eye.
Accommodative responses of the right eye were measured by an autorefractor (Grand Seiko WV-500; Grand Seiko, Tokyo, Japan). An infrared light source in the device projects light into the eye, and the light vergence of the reflection from the retina is analyzed to determine the refractive state. The right eye could not see the experimental stimuli due to a diffusing film that scattered the light. Thus, that eye saw a uniform field of roughly the same luminance as the stimuli presented to the left eye. The uniform field served the purpose of making the infrared light from the autorefractor invisible to the right eye. 
In its normal mode of operation, the autorefractor's sampling rate is ∼1 Hz. However, offline analysis of the composite video signal provided a faster rate of 30 Hz (Cholewiak, Love, Srinivasan, Ng, & Banks, 2017; MacKenzie, Hoffman, & Watt, 2010; Wolffsohn, O'Donnell, Charman, & Gilmartin, 2004). Data corrupted by blinks were discarded. One of course wants to know the magnitude of the noise in the measurements. To estimate this, we made measurements with a stimulus fixed at 3.0 D in two subjects. The standard deviation of the resulting measurements was 0.14 D, much smaller than the effects we describe in the results. 
Stimuli and procedure
The experimental stimuli were black-and-white textured planes. The textures had broadband amplitude spectra of ∼1/f, where f is spatial frequency. There were four precomputed textures that had similar luminance, contrast energy, and amplitude spectra. On each trial, one of the four was randomly chosen for presentation. Each trial began when the subject pressed a start key. A sharp texture at 3.0 D (33 cm) appeared for 3 s with a Maltese cross in the center. Subjects were told to fixate and maintain focus on the cross throughout a trial. Then the experimental stimulus appeared for 10 s. The real or simulated focal distance of the fronto-parallel textured plane oscillated sinusoidally from 1.5–4.5 D. The frequency of oscillation was 0.1, 0.2, 0.5, or 1 Hz. The stimulus had a central circular region centered on the fixation target. The central disk was surrounded by either a uniform gray or an abutting textured annulus with an outer diameter of 14°. The central disk had diameters of 1°, 2°, 4°, 6°, 8°, or 14° (full field). At the end of the 10 s presentation of the experimental stimulus, the texture was replaced with a uniform gray field with the Maltese cross in the center. This remained on for at least 2 s. The next trial began with the subject's key press. Accommodative responses could trigger vergence responses, so the experimenter had to continuously align the autorefractor with the subject's right eye in order to maintain measurement of accommodative state. Vergence responses were slow and predictable, so it was not difficult to maintain alignment. Nevertheless, the procedure may have occasionally resulted in off-axis measurement. For a subject who converges precisely to the change in stimulus distance, vergence to the minimum (1.5 D) and maximum (4.5 D) distances would be 5.4° and 16.1°. However, the eyes would not have converged more than 10° because the accommodative stimulus was the only drive for vergence and the gain of accommodative vergence is less than 1 (Sweeney, Seidel, Day, & Gray, 2014). Wolffsohn et al. (2004) reported errors less than 0.25 D for a similar measurement procedure when measuring 10° off-axis. Therefore, off-axis measurement errors could have a measurable effect, but they would be much smaller than the responses we observed. 
Experimental conditions
There were three experimental conditions: real change, simulated change, no conflict, and simulated change, conflict. These conditions are illustrated in Figure 3 along with still image captures of the stimuli when the diameter of the central disk was 6°. 
Figure 3
 
Experimental conditions. There were three conditions: real change, simulated change, no conflict, and simulated change, conflict. In the real change condition, a central textured disk was surrounded by a uniform gray annulus with an outer diameter of 14°. The diameter of the central disk varied from trial to trial. The optical distance of the stimulus varied sinusoidally due to changes in the power of the focus-adjustable lens. The image at the bottom schematizes this condition. The stimulus appeared initially at 3.0 D (33 cm, darker gray) and then the power of the focus-adjustable lens was changed so the stimulus appeared at a different optical distance (lighter gray). In simulated change, no conflict, a central textured disk of various diameters was surrounded by a uniform gray annulus. The optical distance of the disk was always 3.0 D, but its simulated distance varied sinusoidally by appropriate blurring of the R, G, and B primaries (described in the text). The image at the bottom shows that the optical distance did not change. In simulated change, conflict, a central textured disk of various diameters was surrounded by a peripheral textured annulus. The optical distances of the disk and annulus were always 3.0 D. The simulated distances of the disk and annulus oscillated in counter-phase by appropriate blurring of the R, G, and B primaries such that when the central disk approached, the peripheral annulus receded, and so forth. The image at the bottom shows that the optical distance did not change as the simulated distances of the disk and annulus oscillated in counter-phase.
Figure 3
 
Experimental conditions. There were three conditions: real change, simulated change, no conflict, and simulated change, conflict. In the real change condition, a central textured disk was surrounded by a uniform gray annulus with an outer diameter of 14°. The diameter of the central disk varied from trial to trial. The optical distance of the stimulus varied sinusoidally due to changes in the power of the focus-adjustable lens. The image at the bottom schematizes this condition. The stimulus appeared initially at 3.0 D (33 cm, darker gray) and then the power of the focus-adjustable lens was changed so the stimulus appeared at a different optical distance (lighter gray). In simulated change, no conflict, a central textured disk of various diameters was surrounded by a uniform gray annulus. The optical distance of the disk was always 3.0 D, but its simulated distance varied sinusoidally by appropriate blurring of the R, G, and B primaries (described in the text). The image at the bottom shows that the optical distance did not change. In simulated change, conflict, a central textured disk of various diameters was surrounded by a peripheral textured annulus. The optical distances of the disk and annulus were always 3.0 D. The simulated distances of the disk and annulus oscillated in counter-phase by appropriate blurring of the R, G, and B primaries such that when the central disk approached, the peripheral annulus receded, and so forth. The image at the bottom shows that the optical distance did not change as the simulated distances of the disk and annulus oscillated in counter-phase.
In the real change condition, a textured disk, centered on the fixation target, was presented along with a uniform gray surround. The texture was always rendered sharp. The uniform gray provided no useful information for accommodation. Blur was created by altering the power of the focus-adjustable lens, thereby changing the optical distance of the stimulus. Thus, this condition reproduced all blur cues (defocus, chromatic aberration, higher-order aberrations) commensurate with changes in the distance of an oscillating plane. However, the size of the image did not change because we wished to investigate blur-driven accommodation. 
In the simulated change, no conflict condition, a central textured disk was again presented along with a uniform gray surround. The optical distance of the stimulus was always 3.0 D (33 cm). The size of the image never changed as a function of the simulated distance. Simulated distance changed according to a color-correct rendering method: ChromaBlur (Cholewiak et al., 2017; Cholewiak et al., 2018). The R, G, and B color channels of the image were differentially blurred according to the longitudinal chromatic aberration of the human eye (Thibos, Ye, Zhang, & Bradley, 1992). This rendering method robustly drives accommodation to step (Cholewiak et al., 2017) and sinusoidal (Cholewiak et al., 2018) changes in simulated distance. Indeed, response gains and phases were found to be essentially identical to those produced by real changes in optical distance. Figure 4 illustrates an example simulated change, no conflict stimulus. 
 
Figure 4
 
Example video of a 0.2 Hz, 6° stimulus for the simulated change, no conflict condition. The textured stimulus on the left was presented to observers. An aperture has been added to illustrate the view an observer would experience through the eyepiece. The plot on the right shows the simulated distances for the central disk (blue line). The green dashed vertical line labels the current frame when played.
In the simulated change, conflict condition, a central textured disk was presented along with an abutting textured annulus. The optical distance of the disk and annulus was always 3.0 D. Their simulated distances changed according to the ChromaBlur rendering method. The changes in simulated distances were in opposite directions (180° out of phase), so when the distance to the central disk increased, the distance to the peripheral annulus decreased, and so forth. We conducted a short experiment to determine if subjects consciously perceived the opposing changes in the central and peripheral stimuli and found that they did. Figure 5 illustrates an example of simulated change, conflict stimulus. 
 
Figure 5
 
Example video of a 0.2 Hz, 6° stimulus for the simulated change, conflict condition. The textured stimulus on the left was presented to observers. An aperture has been added to illustrate the view an observer would experience through the eyepiece. The plot on the right shows the simulated distances for the 6° central disk (blue line) and the 6°–14° peripheral annulus (red line). The green dashed vertical line labels the current frame when played.
A variety of cues were available to drive accommodative responses in the real change condition: defocus blur, chromatic aberration, higher-order aberrations, and microfluctuations of accommodation. But in our simulated change conditions, only defocus blur and chromatic aberration could drive responses; higher-order aberrations and microfluctuations indicated that no change in distance had occurred. The fact that we observed responses in the simulated change conditions that were nearly as robust as those in the real change condition shows that defocus blur and chromatic aberration are quite effective in driving accommodation. 
In preliminary testing, we measured each subject's pupil diameter while viewing the stimuli. They were 3.4–5.1 mm with a mean of 4.2 mm. We used diameters of 4 or 5 mm depending on the subject for computing the stimuli in the two simulated change conditions (Cholewiak et al., 2018). 
Each subject completed 360 trials: three experimental conditions, six stimulus diameters, four temporal frequencies, and five repetitions of each. The conditions were randomly interleaved. 
Results
Figure 6 shows all accommodative responses in all conditions for one subject. (The data from the other subjects are provided in Supplementary Figures S1S11; age and refractive error of each subject are also provided.) The colored traces are the responses on every trial: red for real change, blue for simulated, no conflict, and green for simulated, conflict conditions. The dashed black curves represent the stimulus. We fit the raw data from each trial with a sinusoid of the same frequency as the driving stimulus. Fitting parameters were amplitude, phase, and offset. The solid black curves in the figure are the best average fits for that subject. For each trial, we calculated the response gain (the amplitude of the fitted sinewave divided by the amplitude of the driving sinewave) and the response phase (the phase of the fitted sinewave minus the phase of the driving sinewave). A gain of 1 would indicate that the response amplitude was equal to the stimulus amplitude. A phase of 0° would mean that there was no time lag of the response relative to the stimulus. 
Figure 6
 
One subject's accommodative responses in all experimental conditions. In each panel, response in diopters is plotted against time in seconds. The stimulus to accommodation always oscillated around a central value of 3.0 D. Each contains five traces, one for each of the five repetitions of each condition. Columns from left to right show the results when the central disk diameter was 14°, 8°, 6°, 4°, 2°, and 1°, respectively. The upper, middle, and lower parts show the results from the real change (red traces), simulated change, no conflict (blue), and simulated change, conflict (green) conditions, respectively. For each of those conditions, rows from top to bottom show the results for stimulus frequencies of 0.1, 0.2, 0.5, and 1.0 Hz, respectively. The black dashed curves are the stimulus. The black solid curves are the average sinusoids that best-fit the five sets of data in each panel.
Figure 6
 
One subject's accommodative responses in all experimental conditions. In each panel, response in diopters is plotted against time in seconds. The stimulus to accommodation always oscillated around a central value of 3.0 D. Each contains five traces, one for each of the five repetitions of each condition. Columns from left to right show the results when the central disk diameter was 14°, 8°, 6°, 4°, 2°, and 1°, respectively. The upper, middle, and lower parts show the results from the real change (red traces), simulated change, no conflict (blue), and simulated change, conflict (green) conditions, respectively. For each of those conditions, rows from top to bottom show the results for stimulus frequencies of 0.1, 0.2, 0.5, and 1.0 Hz, respectively. The black dashed curves are the stimulus. The black solid curves are the average sinusoids that best-fit the five sets of data in each panel.
Response gains for the real change and simulated, no conflict conditions were high at 0.1 Hz (gain = 0.78) and 0.2 Hz (0.57), somewhat low at 0.5 Hz (0.30), and quite low at 1.0 Hz (0.20). For this reason, the effects of foveal and nonfoveal stimulation are more evident at the two lower frequencies. When the central disk was 14°, responses in all conditions were locked to the stimulus as it oscillated back and forth. This makes sense because with that diameter the visual field was filled with just one oscillating stimulus. As the central disk became smaller, there was little effect on response gain or phase in the real change and simulated, no conflict conditions. This too makes sense because there was no competing peripheral stimulus in those conditions. There was, however, a clear effect of decreasing disk diameter in the simulated, conflict condition. Recall that in that condition the peripheral annulus oscillated in counter-phase to the central disk. As the central disk became smaller and the peripheral annulus larger, the annulus had more and more influence on accommodative responses. Look, for example, at the responses in the simulated, no conflict and simulated, conflict conditions when the central disk diameter was 1°. Here the responses in the conflict condition were clearly shifted by 180° relative to the responses in the no-conflict condition. In other words, the peripheral annulus became the primary driver of accommodative responses when the central disk's diameter was small. 
To quantify the relative influences of the central and peripheral stimuli, we converted the data into polar coordinates as in Figure 7. We first phase-normalized the data to remove the temporal delay that is inherent to accommodative responses. We did this by calculating the phase delay (θ0) in the real change condition for each temporal frequency and subject. We then subtracted that delay from the phase of each of the sinusoids fitted to the trial-by-trial data for the same subject and temporal frequency, but for all three experimental conditions and all six disk diameters:  
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}\tag{1}{\theta _n} = \theta - {\theta _0}\end{equation}
where θ is the phase of the sinusoidal fit on a trial, θ0 is the phase of the median fit across stimulus diameters in the real change condition for the appropriate subject and temporal frequency, and θn is the normalized phase on that trial. This procedure yielded the phase-normalized polar plots of Figure 8. The red, green, and blue vectors are the normalized results for real change, simulated, no conflict, and simulated, conflict, respectively. There are 11 vectors for each condition, temporal frequency, and disk diameter, each representing one subject's average response. Again the gains were high at 0.1 and 0.2 Hz and low at 0.5 and 1.0 Hz, so effects of central and peripheral stimulation are more evident at the lower frequencies. In the real change and simulated, no conflict conditions, the vectors hover around 0°, which means that the responses were, as expected, driven by the central driving stimulus. The responses in the simulated, conflict condition were quite different: They underwent a phase reversal as the central stimulus became smaller. For example, with diameters of 1° and 2°, the response phase was ∼180°, which shows that the responses were more determined by the peripheral stimulus than by the central.  
Figure 7
 
Plotting the results in polar coordinates. The red vector indicates the response on a given trial. Vector length is gain (g), which is the amplitude of the fitted sinusoid divided by the amplitude of the driving stimulus. Vector angle is phase (θ), which is the delay of the fitted sinusoid relative to the driving stimulus. The thick circle represents a gain of 1. The horizontal axis represents a phase of 0°.
Figure 7
 
Plotting the results in polar coordinates. The red vector indicates the response on a given trial. Vector length is gain (g), which is the amplitude of the fitted sinusoid divided by the amplitude of the driving stimulus. Vector angle is phase (θ), which is the delay of the fitted sinusoid relative to the driving stimulus. The thick circle represents a gain of 1. The horizontal axis represents a phase of 0°.
Figure 8
 
Normalized responses. Each panel plots in polar coordinates phase-normalized responses for the 11 subjects: red for real change, blue for simulated change, no conflict, and green for simulated change, conflict. To phase-normalize, we found the average time lag across diameters for each subject and temporal frequency in the real change condition. We then computed the phase for that average lag and subtracted it from all the data for that subject at that frequency in all experimental conditions (Equation 1). The darker circles represent a gain of 1. A few of the vector lengths are slightly greater than 1; they are plotted as lines without a dot at the tip.
Figure 8
 
Normalized responses. Each panel plots in polar coordinates phase-normalized responses for the 11 subjects: red for real change, blue for simulated change, no conflict, and green for simulated change, conflict. To phase-normalize, we found the average time lag across diameters for each subject and temporal frequency in the real change condition. We then computed the phase for that average lag and subtracted it from all the data for that subject at that frequency in all experimental conditions (Equation 1). The darker circles represent a gain of 1. A few of the vector lengths are slightly greater than 1; they are plotted as lines without a dot at the tip.
We decomposed these results into sine (y) and cosine (x) components:  
\begin{equation}\tag{2}x = g{\rm{\ cos}}\left( {{\theta _n}} \right)\end{equation}
 
\begin{equation}\tag{3}y = g{\rm{\ sin}}\left( {{\theta _n}} \right)\end{equation}
where g and θn are gain and normalized phase, respectively. Figure 9 plots the resulting sine and cosine components of the responses, averaged across subjects, as a function of the diameter of the central disk. The upper and lower rows show the sine and cosine components, respectively. Sine and cosine components for individual observers are presented in Supplementary Material Figures S12 and S13. The left, middle, and right columns show them for real change, simulated, no conflict, and simulated, conflict conditions. Red, orange, light blue, and dark blue symbols represent the data for temporal frequencies of 0.1, 0.2, 0.5, and 1.0 Hz, respectively. Again the effects of central and peripheral stimulation are more evident at the two lower frequencies because the gains were higher at those frequencies than at 0.5 and 1 Hz. The noise floor of the fitting procedure was calculated using measurements from two subjects' eyes accommodated to a static Maltese cross at 3.0 D. The means of the noise floor were very close to 0 and 95% confidence intervals were small for the cosine components (0.1 Hz: g = 0.024, 95% CI = [−0.015, 0.061]; 0.5 Hz: 0.004 [−0.011, 0.018]; 0.2 Hz: 0.009 [−0.016, 0.034]; 1.0 Hz: −0.011 [−0.022, −0.001]) and sine components (0.1 Hz: g = −0.020, CI = [−0.080, 0.016]; 0.5 Hz: −0.000 [−0.014, 0.014]; 0.2 Hz: −0.005 [−0.030, 0.016]; 1.0 Hz: 0.007 [−0.001, 0.015]). The main effects we report here are much larger so noise did not affect our conclusions.  
Figure 9
 
Sine and cosine components of responses calculated using Equations 2 and 3. The sine and cosine components averaged across subjects are plotted as a function of the diameter of the central disk. Different colors represent different temporal frequencies as indicated by the legend in the upper right. The sine components are plotted in the top row and cosine components in the bottom row. The left, middle, and right columns show those components for the real change, simulated change, no conflict, and simulated change, conflict conditions, respectively. Error bars are 95% confidence intervals.
Figure 9
 
Sine and cosine components of responses calculated using Equations 2 and 3. The sine and cosine components averaged across subjects are plotted as a function of the diameter of the central disk. Different colors represent different temporal frequencies as indicated by the legend in the upper right. The sine components are plotted in the top row and cosine components in the bottom row. The left, middle, and right columns show those components for the real change, simulated change, no conflict, and simulated change, conflict conditions, respectively. Error bars are 95% confidence intervals.
The sine components were essentially zero for all conditions, central disk diameters, and temporal frequencies. This means that the phase normalization was effective in constraining the data to phases of approximately 0° and 180°. We discuss the implications of this observation in the Combining central and peripheral stimulation section
The cosine components varied systematically across frequencies, conditions, and diameters. In particular, their magnitudes decreased with increasing temporal frequency, which is a manifestation of the decrease in gain seen in Figures 6 and 8. The cosine component did not vary systematically with disk diameter in the real change and simulated, no conflict conditions because there was no competing peripheral stimulus in those conditions. But they did vary with disk diameter in the simulated, conflict condition: As the central disk became smaller, the cosine components became systematically smaller meaning that the peripheral annulus had an increasing influence on the response. This is most obvious at lower temporal frequencies where response gain was high. 
We conducted a repeated-measures ANOVA to assess the statistical reliability of various trends in the data. The results revealed that there was no significant effect of disk diameter in the sine components, F(2, 108) = 1.36; p = 0.24, and that there was a highly significant effect of diameter in the cosine components, F(2, 108) = 4.15; p < 0.0001. The cosine effect was larger at lower temporal frequencies (diameter × frequency interaction: p < 0.0001). The effect of greatest interest is the change in cosine in the simulated, conflict condition as a function of disk diameter. We compared the cosine components at various diameters to the components when the diameter was 14°. At 0.1 Hz, the cosine components were significantly smaller when the disk diameter was 1°, 2°, 4°, 6°, or 8° compared to 14°. At 0.2 Hz, they were significantly smaller when the diameter was 1°, 2°, or 4°. At 0.5 Hz, they were significantly smaller only when the diameter was 1°. At 1.0 Hz, they were not significantly smaller for any diameter compared to 14°. 
In summary, our results demonstrate a surprisingly large influence of nonfoveal stimulation on accommodation even when the fovea is stimulated. For example, there was a significant effect of stimulating nonfoveal retina with an annulus with an inner diameter of 8°. This means that a perifoveal stimulus affects accommodation even when the fovea and parafovea are stimulated with an opposing accommodative stimulus. 
Discussion
We found that stimulation of the perifovea affects accommodation even when the fovea and parafovea are stimulated. Unfortunately, our apparatus did not allow presentation of a stimulus with a diameter greater than 14°, so we do not know how far away from the fovea one can go and still observe an influence of peripheral stimulation on accommodative responses. The observation of a significant influence of the perifovea is surprising because for that part of the retina to affect accommodation it must extract the chromatic and blur signals associated with our simulated blur stimuli and we know that sensitivity to chromatic and blur signals is much reduced in the perifovea (Hansen, Pracejus, & Gegenfurtner, 2009; Vakrou, Whitaker, McGraw, & McKeefry, 2005; Wang & Ciuffreda, 2004). Nonetheless, our results show that stimulation of nonfoveal retina affects accommodation even when the subject is trying to maintain focus on a foveal stimulus. This observation is related to previous work showing that a near occluding object (like pillars in an automobile window) or an aperture (like that in a microscope) can likewise adversely affect the ability to accommodate to a distant target (Chong & Triggs, 1989; Hennessy, 1975). Similarly, a near window screen can disrupt accommodation to a distant stimulus (the Mandelbaum effect; Owens, 1979). 
Combining central and peripheral stimulation
Accommodation is brought about by changes in the power of the eye's crystalline lens. The power changes are instantiated by signals from the motor neurons in the Edinger-Westphal nucleus which activate the ciliary musculature attached to the lens (Kozicz et al., 2011). Only one command can be sent at a time, so sensory signals from different parts of the retina must somehow be combined before the motor command from the nucleus is issued. Here we consider how the signals are combined. 
The key condition in our experiment is the simulated, conflict condition. In that condition, the central and peripheral accommodative stimuli, Sc and Sp, are:  
\begin{equation}\tag{4}{S_c} = a\,{\rm{cos}}(2\pi ft)\end{equation}
 
\begin{equation}\tag{5}{S_p} = a\,{\rm{cos}}(2\pi ft + \pi )\end{equation}
where f and t are temporal frequency and time and a is stimulus amplitude, which was always 1.5 for the central and peripheral stimuli. (In the experiment, the stimulus ranged from +1.5 D to +4.5 D, but to simplify the equations we subtracted 3.0 D so the range of distances was represented as −1.5 to +1.5 D.) The phase of Sp is π, meaning in counter-phase relative to Sc. The two stimuli are then processed through functions C and P producing internal responses Rc and Rp. For now we assume that C and P are linear. Therefore, the internal responses are sinusoidal with different gains and phases:  
\begin{equation}\tag{6}{R_c} = C[{S_c}] = {g_c}\,a\,{\rm{cos}}(2\pi ft + {\theta _c})\end{equation}
 
\begin{equation}\tag{7}{R_p} = P[{S_p}] = {g_p}\,a\,{\rm{cos}}(2\pi ft + \pi + {\theta _p})\end{equation}
where gc and gp are the gains and θc and θp are the phases associated with the signals that have passed through C and P. We are not interested in the overall temporal lag in the accommodative response, so as we described earlier, we use the observed responses in the real change condition as estimates of the overall temporal lag and then eliminate the overall lag by subtracting the phases in that condition from the responses in the other conditions yielding:  
\begin{equation}\tag{8}{R_c} = {g_c}\,a\,{\rm{cos}}(2\pi ft)\end{equation}
 
\begin{equation}\tag{9}{R_p} = {g_p}\,a\,{\rm{cos}}(2\pi ft + \pi + {\theta _\delta })\end{equation}
where θδ = θpθc. Hypothetical responses Rc and Rp are represented in polar coordinates in the upper panel of Figure 10. We assume for now that the combined output response is simply the weighted sum of the internal central and peripheral responses:  
\begin{equation}\tag{10}R = {w_c}{R_c} + {w_p}{R_p}\end{equation}
where wc and wp add to 1. We make the simplifying assumption that gc and gp are 1 (alternatively, we can use scalars for central and peripheral processes that incorporate w and g). As before, we can decompose R into its sine and cosine components. We calculated those components for various weights wp and phase offsets θδ. The lower panel of Figure 10 plots the sine components as a function of wp. Of course, wp will increase as the diameter of the central stimulus decreases because the peripheral annulus becomes larger. The colored lines represent the expected components for various values of θδ. When θδ is greater than zero, Rp has greater lag relative to the stimulus than Rc, yielding negative sine components that become increasingly negative as more weight is given to the peripheral stimulus. This yields negative slope as a function of wp. When θδ is less than zero, Rp has less lag than Rc and the sine components become positive with positive slope as a function of wp. The observed sine components in our simulated, conflict condition hovered around zero and did not change as a function of disk diameter (Figure 9, upper right; Supplementary Materials Figure S12, right column). Thus our results clearly indicate that there is no time lag of the peripheral process relative to the central process.  
Figure 10
 
Effect of phase differences between central and peripheral responses. The upper panel shows the parameters in polar coordinates; wc and wp are the relative weights of responses to the central and peripheral stimuli, respectively. We assume that wc + wp = 1; θδ is the phase of the peripheral process relative to central process. The lower panel plots the sine components of the predicted accommodative responses as a function of wp. The colored lines show those components for different values of θδ.
Figure 10
 
Effect of phase differences between central and peripheral responses. The upper panel shows the parameters in polar coordinates; wc and wp are the relative weights of responses to the central and peripheral stimuli, respectively. We assume that wc + wp = 1; θδ is the phase of the peripheral process relative to central process. The lower panel plots the sine components of the predicted accommodative responses as a function of wp. The colored lines show those components for different values of θδ.
We next investigated the linearity of the internal responses used to generate the combined response from the central and peripheral stimuli. We determined the expected combined response if processes C and P are linear or have compressive or expansive nonlinearities. From Equations 8 and 9 and ignoring the gain, we have:  
\begin{equation}\tag{11}{R_c} = C[{S_c}] = {[{\rm{cos}}(2\pi ft)]^n}\end{equation}
 
\begin{equation}\tag{12}{R_p} = C[{S_p}] = {[{\rm{cos}}(2\pi ft + {\theta _\delta })]^n}\end{equation}
 
Note that n = 1 is a linear process and n less than or greater than 1 are compressive and expansive nonlinearities, respectively. Figure 11 shows the combined responses for linear and nonlinear processes when the weights assigned to the central and peripheral processes are equal or unequal. The red and blue traces in each panel represent the internal responses Rp and Rc and the black traces the output response after summation of the internal responses. The left panels show the internal and the output responses when the weights wp and wc are equal and the right panels show them when they are unequal. The upper, middle, and lower panels show the responses for respectively linear (n = 1), compressive, nonlinear (n = 0.5) and expansive, nonlinear processes (n = 1.5). As you can see, nonlinear processes create energy at 2f, the second harmonic of the driving frequency. The phase of 2f reverses from compressive to expansive nonlinearities; 2f has energy even when the weights of the central and peripheral processes differ. You can also see that linear processes do not yield energy at 2f whether the weights are equal or not. 
Figure 11
 
Linear and nonlinear processes and their effect on the output response. The red and blue curves in each panel represent the internal responses to the peripheral and central stimuli, respectively. The black curves represent the output response after summation of the internal responses. The left panels show these when the peripheral and central stimuli have equal influence on the output: (wc = wp = 0.5) and the right panels show them when they have unequal influence (wc = 0.4; wp = 0.6). The upper, middle, and lower panels show the responses when the central processes are respectively linear (n = 1), have a compressive nonlinearity (n = 0.5), and have an expansive nonlinearity (n = 1.5). The output responses have energy at the second harmonic of the driving frequency in the compressive and expansive cases whether the weights are equal or not. The phase of the second harmonic reverses from compressive to expansive nonlinearity. When the processes are linear and the weights unequal, the output response is not neutralized as it is when the weights are equal, but the output modulation is at the driving frequency and not the second harmonic.
Figure 11
 
Linear and nonlinear processes and their effect on the output response. The red and blue curves in each panel represent the internal responses to the peripheral and central stimuli, respectively. The black curves represent the output response after summation of the internal responses. The left panels show these when the peripheral and central stimuli have equal influence on the output: (wc = wp = 0.5) and the right panels show them when they have unequal influence (wc = 0.4; wp = 0.6). The upper, middle, and lower panels show the responses when the central processes are respectively linear (n = 1), have a compressive nonlinearity (n = 0.5), and have an expansive nonlinearity (n = 1.5). The output responses have energy at the second harmonic of the driving frequency in the compressive and expansive cases whether the weights are equal or not. The phase of the second harmonic reverses from compressive to expansive nonlinearity. When the processes are linear and the weights unequal, the output response is not neutralized as it is when the weights are equal, but the output modulation is at the driving frequency and not the second harmonic.
We measured response modulation in our data set at the driving frequency f (0.1, 0.2, 0.5, and 1 Hz) and the second harmonic 2f. There was modulation at 2f, so we asked whether that modulation was due to a consistent nonlinearity or to random variation. We did this in two ways. First, we compared the modulation at 2f (for example, 0.4 Hz) when the driving frequency was f (in the example, 0.2 Hz) as opposed to when the driver was at a different frequency. Modulation at 2f was not consistently different whether the driver was at f or at another frequency, which suggests that the modulation at 2f was due to random variation. We also examined the phase of modulation at 2f. If the modulation were due to a consistent nonlinearity, the phase of 2f should be consistent for a given driving frequency and experimental condition. In fact, the phase of 2f was essentially random which again suggests that the modulation at 2f was not caused by a consistent nonlinearity. We conclude that the combination of foveal and nonfoveal stimulation to drive an accommodative response is linear or close to linear. 
Foveated rendering
Foveated rendering is used in many displays with a wide field of view (e.g., head-mounted displays). Such rendering produces images with progressively less detail outside the region the viewer is fixating (Guenter, Finch, Drucker, Tan, & Snyder, 2012; Patney et al., 2016). The reduction in detail is meant to be unnoticeable to the viewer because regions with less detail fall on nonfoveal regions where visual resolution is much lower than in the fovea (Rovamo & Virsu, 1979). Foveated rendering reduces computational load and thereby enables significant speed improvements. Of course, eye tracking is required in order to determine which regions of the display should be rendered with full detail and which can be rendered with less detail. Our finding that nonfoveal stimulation affects accommodation should be considered in foveated-rendering algorithms because it is possible that reducing the resolution of nonfoveal stimulation will affect users' accommodation. 
Lags and leads of accommodation
Accommodative responses often exhibit lags (response in diopters less than stimulus; eye under-accommodated) and leads (response greater than stimulus; eye over-accommodated). Our results show that the stimulus on nonfoveal retina affects accommodation even when the fovea is stimulated as well. It is possible that some of these lag and lead errors are caused by an influence of the nonfoveal stimulus. For example, consider an eye in which a well-focused image on the fovea is accompanied by an image focused in front of perifoveal retina. Our findings suggest that the eye will relax accommodation in order to bring the perifoveal image closer to the retina. This will in turn produce an apparent accommodative lag at the fovea. Of course, this hypothesis will have difficulty explaining why both lags and leads can be observed in the same eye. 
Development of myopia
We observed that nonfoveal stimulation affects accommodation even when the viewer is fixating a foveal stimulus. This occurs despite the fact that the ability to extract blur and chromatic signals is poorer in nonfoveal regions than in the fovea (Hansen et al., 2009; Wang & Ciuffreda, 2004). Our observation may help us understand how images formed relative to nonfoveal retina can affect eye growth (Smith et al., 2005; Smith III et al., 2009). Specifically, images that are habitually focused behind the nonfoveal retina may cause an increase in accommodation, which in turn promotes eye growth (Ebenholtz, 1983) even when the foveal image is in focus. Over time the eye lengthening produces myopia. 
Acknowledgments
This work was funded by NSF Research Grant BCS-1734677 and by Corporate University Research, Intel Labs. The authors thank Mike Landy for discussing the sine and cosine analysis. Also, this open access publication was made possible in part by support from the Berkeley Research Impact Initiative (BRII) sponsored by the UC Berkeley Library. 
Commercial relationships: MSB & SAC patent pending #PCT/US2017/031117. 
Corresponding author: Vivek Labhishetty. 
Address: Optometry & Vision Science, University of California, Berkeley, CA, USA. 
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Figure 1
 
Stimulation of foveal and nonfoveal retina. A smart phone is held at a distance of 50 cm and the viewer fixates the center of it. The concentric rings represent the foveola, fovea, parafovea, and perifovea, and how they would be stimulated in this situation. The perifovea is stimulated by more distant points than the fovea.
Figure 1
 
Stimulation of foveal and nonfoveal retina. A smart phone is held at a distance of 50 cm and the viewer fixates the center of it. The concentric rings represent the foveola, fovea, parafovea, and perifovea, and how they would be stimulated in this situation. The perifovea is stimulated by more distant points than the fovea.
Figure 2
 
Experimental apparatus. Stimuli were projected from a DLP projector through a triple-bandpass filter onto a screen. The stimuli were viewed by the subject's left eye through a focus-adjustable lens, fixed offset lens, and aperture. The right eye's refractive state was measured by an autorefractor. Infrared light was shone into that eye and the vergence of its reflection from the retina was analyzed. A diffusing screen scattered the visible light from the experimental stimulus creating a uniform field seen by the right eye. A hot mirror enabled transmission of visible light to the two eyes and reflection of infrared light to the right eye.
Figure 2
 
Experimental apparatus. Stimuli were projected from a DLP projector through a triple-bandpass filter onto a screen. The stimuli were viewed by the subject's left eye through a focus-adjustable lens, fixed offset lens, and aperture. The right eye's refractive state was measured by an autorefractor. Infrared light was shone into that eye and the vergence of its reflection from the retina was analyzed. A diffusing screen scattered the visible light from the experimental stimulus creating a uniform field seen by the right eye. A hot mirror enabled transmission of visible light to the two eyes and reflection of infrared light to the right eye.
Figure 3
 
Experimental conditions. There were three conditions: real change, simulated change, no conflict, and simulated change, conflict. In the real change condition, a central textured disk was surrounded by a uniform gray annulus with an outer diameter of 14°. The diameter of the central disk varied from trial to trial. The optical distance of the stimulus varied sinusoidally due to changes in the power of the focus-adjustable lens. The image at the bottom schematizes this condition. The stimulus appeared initially at 3.0 D (33 cm, darker gray) and then the power of the focus-adjustable lens was changed so the stimulus appeared at a different optical distance (lighter gray). In simulated change, no conflict, a central textured disk of various diameters was surrounded by a uniform gray annulus. The optical distance of the disk was always 3.0 D, but its simulated distance varied sinusoidally by appropriate blurring of the R, G, and B primaries (described in the text). The image at the bottom shows that the optical distance did not change. In simulated change, conflict, a central textured disk of various diameters was surrounded by a peripheral textured annulus. The optical distances of the disk and annulus were always 3.0 D. The simulated distances of the disk and annulus oscillated in counter-phase by appropriate blurring of the R, G, and B primaries such that when the central disk approached, the peripheral annulus receded, and so forth. The image at the bottom shows that the optical distance did not change as the simulated distances of the disk and annulus oscillated in counter-phase.
Figure 3
 
Experimental conditions. There were three conditions: real change, simulated change, no conflict, and simulated change, conflict. In the real change condition, a central textured disk was surrounded by a uniform gray annulus with an outer diameter of 14°. The diameter of the central disk varied from trial to trial. The optical distance of the stimulus varied sinusoidally due to changes in the power of the focus-adjustable lens. The image at the bottom schematizes this condition. The stimulus appeared initially at 3.0 D (33 cm, darker gray) and then the power of the focus-adjustable lens was changed so the stimulus appeared at a different optical distance (lighter gray). In simulated change, no conflict, a central textured disk of various diameters was surrounded by a uniform gray annulus. The optical distance of the disk was always 3.0 D, but its simulated distance varied sinusoidally by appropriate blurring of the R, G, and B primaries (described in the text). The image at the bottom shows that the optical distance did not change. In simulated change, conflict, a central textured disk of various diameters was surrounded by a peripheral textured annulus. The optical distances of the disk and annulus were always 3.0 D. The simulated distances of the disk and annulus oscillated in counter-phase by appropriate blurring of the R, G, and B primaries such that when the central disk approached, the peripheral annulus receded, and so forth. The image at the bottom shows that the optical distance did not change as the simulated distances of the disk and annulus oscillated in counter-phase.
Figure 6
 
One subject's accommodative responses in all experimental conditions. In each panel, response in diopters is plotted against time in seconds. The stimulus to accommodation always oscillated around a central value of 3.0 D. Each contains five traces, one for each of the five repetitions of each condition. Columns from left to right show the results when the central disk diameter was 14°, 8°, 6°, 4°, 2°, and 1°, respectively. The upper, middle, and lower parts show the results from the real change (red traces), simulated change, no conflict (blue), and simulated change, conflict (green) conditions, respectively. For each of those conditions, rows from top to bottom show the results for stimulus frequencies of 0.1, 0.2, 0.5, and 1.0 Hz, respectively. The black dashed curves are the stimulus. The black solid curves are the average sinusoids that best-fit the five sets of data in each panel.
Figure 6
 
One subject's accommodative responses in all experimental conditions. In each panel, response in diopters is plotted against time in seconds. The stimulus to accommodation always oscillated around a central value of 3.0 D. Each contains five traces, one for each of the five repetitions of each condition. Columns from left to right show the results when the central disk diameter was 14°, 8°, 6°, 4°, 2°, and 1°, respectively. The upper, middle, and lower parts show the results from the real change (red traces), simulated change, no conflict (blue), and simulated change, conflict (green) conditions, respectively. For each of those conditions, rows from top to bottom show the results for stimulus frequencies of 0.1, 0.2, 0.5, and 1.0 Hz, respectively. The black dashed curves are the stimulus. The black solid curves are the average sinusoids that best-fit the five sets of data in each panel.
Figure 7
 
Plotting the results in polar coordinates. The red vector indicates the response on a given trial. Vector length is gain (g), which is the amplitude of the fitted sinusoid divided by the amplitude of the driving stimulus. Vector angle is phase (θ), which is the delay of the fitted sinusoid relative to the driving stimulus. The thick circle represents a gain of 1. The horizontal axis represents a phase of 0°.
Figure 7
 
Plotting the results in polar coordinates. The red vector indicates the response on a given trial. Vector length is gain (g), which is the amplitude of the fitted sinusoid divided by the amplitude of the driving stimulus. Vector angle is phase (θ), which is the delay of the fitted sinusoid relative to the driving stimulus. The thick circle represents a gain of 1. The horizontal axis represents a phase of 0°.
Figure 8
 
Normalized responses. Each panel plots in polar coordinates phase-normalized responses for the 11 subjects: red for real change, blue for simulated change, no conflict, and green for simulated change, conflict. To phase-normalize, we found the average time lag across diameters for each subject and temporal frequency in the real change condition. We then computed the phase for that average lag and subtracted it from all the data for that subject at that frequency in all experimental conditions (Equation 1). The darker circles represent a gain of 1. A few of the vector lengths are slightly greater than 1; they are plotted as lines without a dot at the tip.
Figure 8
 
Normalized responses. Each panel plots in polar coordinates phase-normalized responses for the 11 subjects: red for real change, blue for simulated change, no conflict, and green for simulated change, conflict. To phase-normalize, we found the average time lag across diameters for each subject and temporal frequency in the real change condition. We then computed the phase for that average lag and subtracted it from all the data for that subject at that frequency in all experimental conditions (Equation 1). The darker circles represent a gain of 1. A few of the vector lengths are slightly greater than 1; they are plotted as lines without a dot at the tip.
Figure 9
 
Sine and cosine components of responses calculated using Equations 2 and 3. The sine and cosine components averaged across subjects are plotted as a function of the diameter of the central disk. Different colors represent different temporal frequencies as indicated by the legend in the upper right. The sine components are plotted in the top row and cosine components in the bottom row. The left, middle, and right columns show those components for the real change, simulated change, no conflict, and simulated change, conflict conditions, respectively. Error bars are 95% confidence intervals.
Figure 9
 
Sine and cosine components of responses calculated using Equations 2 and 3. The sine and cosine components averaged across subjects are plotted as a function of the diameter of the central disk. Different colors represent different temporal frequencies as indicated by the legend in the upper right. The sine components are plotted in the top row and cosine components in the bottom row. The left, middle, and right columns show those components for the real change, simulated change, no conflict, and simulated change, conflict conditions, respectively. Error bars are 95% confidence intervals.
Figure 10
 
Effect of phase differences between central and peripheral responses. The upper panel shows the parameters in polar coordinates; wc and wp are the relative weights of responses to the central and peripheral stimuli, respectively. We assume that wc + wp = 1; θδ is the phase of the peripheral process relative to central process. The lower panel plots the sine components of the predicted accommodative responses as a function of wp. The colored lines show those components for different values of θδ.
Figure 10
 
Effect of phase differences between central and peripheral responses. The upper panel shows the parameters in polar coordinates; wc and wp are the relative weights of responses to the central and peripheral stimuli, respectively. We assume that wc + wp = 1; θδ is the phase of the peripheral process relative to central process. The lower panel plots the sine components of the predicted accommodative responses as a function of wp. The colored lines show those components for different values of θδ.
Figure 11
 
Linear and nonlinear processes and their effect on the output response. The red and blue curves in each panel represent the internal responses to the peripheral and central stimuli, respectively. The black curves represent the output response after summation of the internal responses. The left panels show these when the peripheral and central stimuli have equal influence on the output: (wc = wp = 0.5) and the right panels show them when they have unequal influence (wc = 0.4; wp = 0.6). The upper, middle, and lower panels show the responses when the central processes are respectively linear (n = 1), have a compressive nonlinearity (n = 0.5), and have an expansive nonlinearity (n = 1.5). The output responses have energy at the second harmonic of the driving frequency in the compressive and expansive cases whether the weights are equal or not. The phase of the second harmonic reverses from compressive to expansive nonlinearity. When the processes are linear and the weights unequal, the output response is not neutralized as it is when the weights are equal, but the output modulation is at the driving frequency and not the second harmonic.
Figure 11
 
Linear and nonlinear processes and their effect on the output response. The red and blue curves in each panel represent the internal responses to the peripheral and central stimuli, respectively. The black curves represent the output response after summation of the internal responses. The left panels show these when the peripheral and central stimuli have equal influence on the output: (wc = wp = 0.5) and the right panels show them when they have unequal influence (wc = 0.4; wp = 0.6). The upper, middle, and lower panels show the responses when the central processes are respectively linear (n = 1), have a compressive nonlinearity (n = 0.5), and have an expansive nonlinearity (n = 1.5). The output responses have energy at the second harmonic of the driving frequency in the compressive and expansive cases whether the weights are equal or not. The phase of the second harmonic reverses from compressive to expansive nonlinearity. When the processes are linear and the weights unequal, the output response is not neutralized as it is when the weights are equal, but the output modulation is at the driving frequency and not the second harmonic.
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